Arc length
Elementary
Think of a curved path, like a winding road. The arc length of the curve is how long it would be if it were straightened out, like measuring the path with a piece of string.
High school
Consider a curve in the plane or in space. The arc length of the curve is its length, as if it were straightened out.
Undergraduate
Consider a parametrized curve $\gamma : [a, b] \to \reals^n$ whose derivative is continuous. The arc length of $\gamma$ is
\[L = \int_a^b \norm{\gamma^{\prime}(t)} \mathrm{d}t.\]Examples
Worked examples
-
$[0, 1]$
: $1$
High school
The arc length of $[0, 1]$, as a straight segment, is its length, $1 - 0 = 1$.